In this paper, an equivalence relation on an ordered Γ-semigroup S is introduced. Using the relation 0-minimal (m, n)-Γ-ideals of S is studied. The results obtained generalize the results on semigroups(without order) and Γ-semigroups(without order).

We introduce the notion of ordered quasi--ideals of regular ordered -semigroups and study the basic properties of ordered quasi--ideals of ordered -semigroups. We also characterize regular ordered -semigroups in terms of their ordered quasi--ideals, ordered right -ideals, and left -ideals. Finally, we have shown that (i) a partially ordered -semigroup is regular if and only if for every ordered bi--ideal , every ordered -ideal , and every ordered quasi--ideal , we have and (ii) a partially ordered -semigroup is regular if and only if for every ordered quasi--ideal , every ordered left -ideal , and every ordered right--ideal , we have that .

Indian mathematics has its deep roots in the Vedas, different from what is known as Vedic Mathematics. Vedic age gave rise to a new era of progress in the field of Science, Technology and Development. The Hindu Scripture Vedas is synonymous with all kinds of original source of knowledge and intellectual wisdom in the universe leading to modern knowlege in modern mathematics. Indian mathematicians made tremendous contributions to the entire world of Mathematics and Science. Decimal number system as well as the invention of zero (0) are among the greatest contributions of Indian mathematicians. The theory of trigonometry, Mathematical Modelling, Algebra, algorithm, modern arithmetic, sine and cosine functions leading to modern trigonometry, Diophantine equation, square root, cube root, negative numbers are also developed by Indian mathematicians. In this review article, the work of some of the renowned Indian mathematicians from Indus Valley civilization and the Vedas to modern times are covered in short with the hope that it may reveal hidden fundamental mathematical ideas as basic ideal tools which may usefully motivate for further research work in every domain of mathematical sciences, natural and applied sciences, engineering and social sciences. Moreover, there are many more remarkable Indian mathematicians who contributed to the origin of mathematical sciences. They have made several general contributions to mathematics that have significantly influenced scientists and mathematicians in the modern times.

In this article, we deal with ordered generalized $\Gamma$-hyperideals in ordered $\Gamma$-semihypergroups. In particular, we study $(m, n)$-regular ordered $\Gamma$-semihypergroups in terms of ordered $(m, n)$-$\Gamma$-hyperideals. Moreover, we obtain some ideal theoretic results in ordered $\Gamma$-semihypergroups.